Higher-derivative superparticle in AdS$_3$ space

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by  Nikolay Kozyrev

ITP-UH-01/16
Higher-derivative superparticle in AdS space
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Nikolay Kozyreva, Sergey Krivonosa and Olaf Lechtenfeldb
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0 a Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia
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n b Institut fu¨r Theoretische Physik and Riemann Center for Geometry and Physics
a Leibniz Universita¨t Hannover, Appelstrasse 2, D-30167 Hannover, Germany
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nkozyrev, [email protected], [email protected]

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t
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Abstract
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Employing the coset approach we construct component actions for a superparticle moving in AdS3 with
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N=(2,0), D=3 supersymmetry partially broken to N=2, d=1. These actions may contain higher time-
0
derivativeterms,whicharechosentopossessthesame(super)symmetriesasthefreesuperparticle. Intermsof
.
1 thenonlinear-realization superfields,thecomponentactionsalwaystakeasimplerformwhenwritteninterms
0
of covariant Cartan forms. We also consider in detail the reduction to the nonrelativistic case and construct
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the corresponding action a Newton-Hooke superparticle and its higher-derivative generalizations. The struc-
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tureofthesehighertime-derivativegeneralizationsiscompletelyfixedbyinvarianceunderthesupersymmetric
:
v Newton-Hooke algebra extendedby two central charges.
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a
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1 Introduction
The standardactionofany particlemovingina flatspacetime is invariantunderthe target-spacePoincar´egroup
realized in a spontaneously broken manner. The spontaneously broken translations, orthogonal to the world-line
of particle, and the Lorentz boosts rotating these translations into world-line translations, give rise to Goldstone
bosons, which appear in the particle actions. Usually, not all of these Goldstone bosons are independent of one
another,andsothere are additionalconstraintsreducing the number ofindependent fields to a set describingthe
physical degrees of freedom. In the supersymmetric case the situation is more complicated, because in extended
supersymmetry additional covariant constraints selecting irreducible supermultiplets have to be found. These
tasks can be algorithmically solved by using the nonlinear-realization (or coset) approach [1], suitably modified
for supersymmetric spacetime symmetries [2]. In this approach, the corresponding constraints are conditions on
the Cartan forms (or on their θ-components). Nevertheless, the coset approachfails to reproduce the superspace
actions, because the superparticle Lagragian is only quasi-invariant with respect to the super-Poincar´e group
and,therefore,itcannotbeconstructedintermsofCartanforms. However,bypassingtocomponentactionsand
focussingonthe brokensupersymmetryonly,onemayeasilyconstructanansatzforthe invariantactioninterms
of θ = 0 projections of the Cartan forms. This program has been performed in our paper [4] for a superparticle
moving in flat D = 1+2 spacetime with N=4 supersymmetry partially broken to N=2. Moreover, the possible
higher time-derivative terms, possessing the same symmetry as the free superparticle, can also be constructed
in terms of the Cartan forms. The role of unbroken supersymmetry is just to fix some free coefficients in the
component action.
In the presentpaper, we investigate the applicationof the coset approachto a superparticle moving on AdS .
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This leads to the following complications:
Onehastochooseasuitableparametrizationofthecosetspace(i.e.ofthesetofthephysicalbosonicfields),
•
which may, even in the bosonic case, drastically simplify the resulting actions;
The superspace constraints have to be properly covariantized, selecting irreducible supermultiplets which
•
are rather distinct from the flat-spacetime case;
The fermionic components shouldbe defined suchasto renderthe resultingLagrangiansreadable. All such
•
choicesarerelatedbynonlinearinvertiblefieldsredefinitions[5],butanimproperchoicemayresultinhighly
complicated Lagrangians.
In the following we will solve all these tasks for a superparticle moving on AdS , with N=(2,0), D=3 supersym-
3
metry partially broken to N=2, d=1. We will construct the corresponding component actions in terms of the
θ = 0 projections of the Cartan forms and prove their invariance under the N=(2,0) AdS algebra. The higher
3
time-derivative terms share this symmetry, and the one with minimal higher derivatives (the anyonic action) can
also be written in terms of Cartan forms. The analysis will then be extended by the nonrelativistic limit, which
radically simplifies everything: The AdS superparticle reduces to the Newton-Hooke superparticle [6, 15], while
3
the higher derivative terms acquire quite a compact form.
2 N = (2,0) AdS algebra and fixing the basis
3
The action we are going to construct corresponds to the partial spontaneous breaking of N = (2,0) AdS su-
3
persymmetry. To start with, let us define the N = (2,0) AdS super algebra in a standard way as (see e.g.
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[3])
[M ,M ]=ǫ M +ǫ M +ǫ M +ǫ M (M) ,
ab cd ac bd bd ac ad bc bc ad ≡ ab,cd
m2
[M ,P ]=(P) , [P ,P ]= (M) ,
ab cd ab,cd ab cd −16 ab,cd
[M ,Q ]=ǫ Q +ǫ Q (Q) , M ,Q = Q ,
ab c ac b bc a ≡ ab,c ab c ab,c
m m
[P ,Q ]=i (Q) , P ,Q =i (cid:2) Q (cid:3), [(cid:0)J,Q(cid:1) ]=Q , J,Q = Q ,
ab c 4 ab,c ab c 4 ab,c a a a − a
m
Q ,Q =2P +i M(cid:2) +im(cid:3)ǫ J. (cid:0) (cid:1) (cid:2) (cid:3) (2.1)
a b ab ab ab
2
(cid:8) (cid:9)
Here, the generators M = M ,P = P , a,b = 1,2 form the bosonic AdS algebra while the fermionic
ab ba ab ba 3
generators Q ,Q together with the U(1) generator J extend it to the N =(2,0) AdS one.
a a 3
1
Note, that in our basis these generators obey the following conjugation rules
(M )† = M , (P )† =P , (J)† =J, (Q )† =Q . (2.2)
ab ab ab ab a a
−
To have close relations with the previously considered case of super particle moving in three-dimensional
Poincar´espace-time [4], one has to choose the generatorsspanning N =2,d=1 super Poincar´ealgebrato which
the AdS supersymmetry will be broken to. One may easily check that if we define the generators P,Q,Q as
3
m (cid:8) (cid:9)
Q=Q +iQ , Q=Q iQ , P =P +P +i (M +M )+mJ, (2.3)
1 2 1 2 11 22 11 22
− 4
then they will form the N =2,d=1 super Poincar´ealgebra
Q,Q =2P, Q,Q = Q,Q =[P,Q]= P,Q =0. (2.4)
{ }
The remaining bosonic gene(cid:8)rators(cid:9),having the proper fo(cid:8)rm in(cid:9)the flat limit,(cid:2)may(cid:3)be defined as follows:
m
P, Z =P P 2iP +i (M M 2iM ), Z =(Z)†,
11 22 12 11 22 12
− − 4 − −
i i
J = (M +M ), T = (M M 2iM ), T =(T)†. (2.5)
3 11 22 11 22 12
4 4 − −
Thus, the bosonic part of the algebra (2.1), i.e. the algebra so(2,2) u(1), acquires the form:
×
[J ,T]=T, J ,T = T, T,T = 2J ,
3 3 3
− −
[P,Z]=2mZ(cid:2), P,(cid:3)Z = 2m(cid:2) Z, (cid:3)Z,Z = 4mP +4m2J,
− −
[J3,Z]=Z, J3(cid:2),Z =(cid:3) Z, (cid:2) (cid:3)
−
[T,P]= Z,(cid:2) T,P(cid:3) =Z, T,Z = 2P +2mJ, T,Z =2P 2mJ. (2.6)
− − −
Clearly, the relations (2.6) are maxim(cid:2)ally(cid:3)similar t(cid:2)o the(cid:3)D = 3 Poincar´e(cid:2)ones w(cid:3)e used in [4] and go to them in
the limit m=0 (with decoupled generator J, of course).
As concerning the fermionic part of N =(2,0) AdS superalgebra (2.1), it is natural to define the generators
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of broken supersymmetry as
S =Q +iQ , S =Q iQ . (2.7)
1 2 1 2
−
Then commutation relations, which include spinor generators, read
Q,Q =2P, S,S =2P 4mJ, Q,S =2Z, Q,S =2Z,
− { }
[(cid:8)Z,Q](cid:9)= 2mS(cid:8), Z,(cid:9)Q =2mS, [Z,S]= 2mQ, Z(cid:8),S (cid:9)=2mQ,
− −
[(cid:2)P,S](cid:3)= 2mS, P,S =2mS, (cid:2) (cid:3) (2.8)
−
[T,Q]= S, T,Q =S, [(cid:2)T,S](cid:3)= Q, T,S =Q,
− −
[J3,Q]=−21Q, J(cid:2)3,Q (cid:3)= 21Q, [J3,S]=−12S(cid:2), J3(cid:3),S = 21S,
[J,Q]=Q,(cid:2) J,Q(cid:3)= Q, [J,S]= S, J,(cid:2)S =(cid:3)S.
− −
(cid:2) (cid:3) (cid:2) (cid:3)
3 Cartan forms and transformation properties
In the cosetapproach[1, 2], the spontaneous breakdownofS,S supersymmetry andZ,Z translationsis reflected
in the structure of the coset element
g =eitPeθQ+θ¯QeψS+ψ¯Sei(UZ+UZ)ei(ΛT+ΛT). (3.1)
The N = 2 superfields U(t,θ,θ¯),ψ(t,θ,θ¯) and Λ(t,θ,θ¯) are Goldstone superfields accompanying the N = (2,0)
AdS symmetry to N =2,d=1 super-Poincar´e U(1)2 breaking1. The transformation properties of the coordi-
3
×
nates and the superfields are induced by the left multiplication of the coset element (3.1)
g g =g′ h, h eiαJeiβJ3.
0
∼
The most important transformations read
1Thesetwoadditional U(1)groupsareformedbythegenerators J andJ3.
2
Unbroken SUSY g =eǫQ+ǫ¯Q
0
•
(cid:16) (cid:17) δθ =ǫ, δθ¯=ǫ¯, δt=i ǫθ¯+¯ǫθ . (3.2)
(cid:0) (cid:1)
Broken SUSY g =eεS+ε¯S
0
•
(cid:16) (cid:17)
δ θ =4mε˜ψ¯θ, δ t=i ε˜ψ¯ +ε¯˜ψ 1 6mθθ¯ 4m ε˜θu ε¯˜θ¯u¯ (1 2mψψ¯), (3.3)
S S
− − − −
δSψ =ε˜ 1 2mθθ¯ 1+(cid:0)2mψψ¯ (cid:1)(cid:0)8im2ψ ε˜θu(cid:1) ε¯˜θ¯u¯(cid:0), δSu=(cid:1)2iε¯˜θ¯ 1 2mψψ¯ 1 4m2uu¯ ,
− − − − −
where ε˜=e2imtε(cid:0). (cid:1)(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:1)
Z,Z-transformations g =ei(bZ+¯bZ)
0
•
(cid:16) (cid:17)
δ θ =2im¯˜bψ 1+2mθθ¯ , δ t=4m ˜bθψ ˜¯bθ¯ψ¯ 2im ˜bu¯ ¯˜bu 1+2mθθ¯ 1 2mψψ¯ ,
Z Z
− − − −
δ u=˜b 1 4(cid:0)m2uu¯ 1(cid:1)+2mθθ¯ 1 (cid:16)2mψψ¯ , (cid:17) (cid:16) (cid:17)(cid:0) (cid:1)(cid:0) (cid:1)
Z
− −
δ ψ =2(cid:0)im¯˜bθ¯ 1+2m(cid:1)ψ(cid:0)ψ¯ +4m(cid:1)2ψ(cid:0) ˜bu¯ ¯˜bu (cid:1) 1+2mθθ¯ . (3.4)
Z
−
(cid:0) (cid:1) (cid:16) (cid:17)(cid:0) (cid:1)
where˜b=e−2imtb.
Here, the coordinates of stereographic projections were introduced
tanh 2m√UU tanh √ΛΛ
u= U, λ= Λ. (3.5)
2(cid:16)m√UU (cid:17) √(cid:16)ΛΛ (cid:17)
The local geometric properties of the system are specified by the left-invariant Cartan forms
g−1dg =iΩ P +iΩ Z+iΩ Z+iΩ T +iΩ T +iΩ J +iΩ J +Ω Q+Ω Q+Ω S+Ω S, (3.6)
P Z Z T T 3 3 J Q Q S S
which look much more complicated than in the flat space-time (m 0) [4]
→
Ω = 1 1+λλ¯ Ωˆ 2i λ¯Ωˆ λΩˆ , Ω = ΩˆQ−iλ¯ΩˆS,
P 1 λλ¯ P − Z − Z Q 1 λλ¯
− h(cid:0) (cid:1) (cid:16) (cid:17)i −
Ω = 1 Ωˆ λ2Ωˆ +iλΩˆ , Ω =Ωˆ 1 2pmλλ¯Ωˆ 2im λ¯Ωˆ λΩˆ ,
Z 1 λλ¯ Z − Z P J J − 1 λλ¯ P − Z − Z
− h i − h (cid:16) (cid:17)i
dλ λdλ¯ dλλ¯ Ωˆ iλ¯Ωˆ
Ω = , Ω =i − , Ω = S − Q, (3.7)
T 1 λλ¯ 3 1 λλ¯ S 1 λλ¯
− − −
where the hatted forms read p
1+4m2uu¯ ˆt+2im(udu¯ u¯du)+8m uψdθ u¯ψ¯dθ¯
Ωˆ = △ − − ,
P 1 4m2uu¯
(cid:0) (cid:1) − (cid:0) (cid:1)
du+2imuˆt 2i ψ¯dθ¯ 4m2u2ψdθ
Ωˆ = △ − − ,
Z 1 4m2uu¯
− (cid:0) (cid:1)
8m3uu¯ u¯du du¯u m2 uψdθ u¯ψ¯dθ¯
Ωˆ = ˆt+2im2 − 8 −
J −1 4m2uu¯△ 1 4m2uu¯ − 1 4m2uu¯
− − (cid:0) − (cid:1)
8m2 dt i θdθ¯+θ¯dθ ψψ¯ +2im ψdψ¯ +ψ¯dψ
− −
= 2m dt(cid:2) i θd(cid:0)θ¯+θ¯dθ (cid:1)(cid:3)mˆt mΩˆ(cid:0)P, (cid:1)
− − △ −
θ 2imu¯ ψ¯ ψ 2imu¯ θ¯
Ωˆ = △ (cid:2)− (cid:0)△ , Ωˆ (cid:1)(cid:3)= △ − △ . (3.8)
Q S
√1 4m2uu¯ √1 4m2uu¯
− −
Here,
ˆt = 1+4mψψ¯ dt i θdθ¯+θ¯dθ+ψdψ¯ +ψ¯dψ 1+4mψψ¯ t,
△ − ≡ △
θ = (cid:0)1 2mψψ¯(cid:1)(cid:2)dθ, (cid:0)ψ =dψ 2imψ dt i θ(cid:1)d(cid:3)θ¯+(cid:0)θ¯dθ . (cid:1) (3.9)
△ − △ − −
(cid:0) (cid:1) (cid:2) (cid:0) (cid:1)(cid:3)
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In what follows, we find it convenient to define the covariant derivatives similarly to the flat case, i.e. with
respect to differentials t,dθ,dθ¯:
△
∂ ∂ ∂
dt +dθ +dθ¯ = t +dθ +dθ¯ . (3.10)
∂t ∂θ ∂θ¯ △ ∇t ∇ ∇
Explicitly, they read
=E−1∂ , =D i ψ¯ ψ+ψ ψ¯ ∂ , =D i ψ¯ ψ+ψ ψ¯ ∂ , (3.11)
t t t t
∇ ∇ − ∇ ∇ ∇ − ∇ ∇
where (cid:0) (cid:1) (cid:0) (cid:1)
∂ ∂
E =1+i ψ˙ψ¯ +ψ¯˙ ψ , D = iθ¯∂ , D = iθ∂ : D,D = 2i∂ . (3.12)
∂θ − t ∂θ¯− t − t
(cid:16) (cid:17) (cid:8) (cid:9)
These derivatives obey the following algebra,
, = 2i 1+ ψ ψ¯ + ψ ψ¯ , , = 4i ψ¯ ψ , , = 4i ψ¯ ψ ,
t t t
∇ ∇ − ∇ ∇ ∇ ∇ ∇ {∇ ∇} − ∇ ∇ ∇ ∇ ∇ − ∇ ∇ ∇
(cid:8)[ t, (cid:9)]= 2i (cid:0) ψ¯ tψ+ ψ tψ¯ (cid:1)t, t, = 2i ψ¯ tψ+ ψ(cid:8) tψ¯ (cid:9)t. (3.13)
∇ ∇ − ∇ ∇ ∇ ∇ ∇ ∇ ∇ − ∇ ∇ ∇ ∇ ∇
(cid:0) (cid:1) (cid:2) (cid:3) (cid:0) (cid:1)
4 Preliminary consideration: the bosonic action
Before considering the full supersymmetric AdS system it makes sense to analyze its bosonic sector.
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The bosonic sector of our N = (2,0) supersymmetric AdS superalgebra (2.1) contains the bosonic AdS
3 3
algebra (i.e. so(2,2) algebra) commuting with the U(1) algebra spanned by the generator J. In this subsection
wearegoingtoconsiderthespontaneousbreakdownofthisAdS S1 symmetrydowntod=1 Poincar´e U(1)2
3
× ×
algebra,generated by P,J and J generators. Therefore, our coset element is just the (θ,ψ 0) limit of the full
3
→
coset element (3.1)
g =eitP ei(UZ+UZ) ei(ΛT+ΛT). (4.1)
The corresponding (θ,ψ 0) limit of the Cartan forms read
→
1 1
ω = 1+λλ¯ ωˆ 2i λ¯ωˆ λωˆ , ω = ωˆ λ2ωˆ +iλωˆ ,
P 1 λλ¯ P − Z − Z Z 1 λλ¯ Z − Z P
− −
(cid:2)(cid:0) (cid:1) (cid:0) (cid:1)(cid:3) (cid:2) (cid:3)
1 dλ λdλ¯ dλλ¯
ω =ωˆ 2mλλ¯ω 2im λ¯ωˆ λωˆ , ω = , ω =i − , (4.2)
J J − 1 λλ¯ P − Z − Z T 1 λλ¯ 3 1 λλ¯
− − −
(cid:2) (cid:0) (cid:1)(cid:3)
where
1
ωˆ = 1+4m2uu¯ dt+2im(udu¯ u¯du) ,
P 1 4m2uu¯ −
−
(cid:2)(cid:0) (cid:1) (cid:3)
1 1
ωˆ = [du+2imudt], ωˆ = [du¯ 2imu¯dt],
Z 1 4m2uu¯ Z 1 4m2uu¯ −
− −
2m2
ωˆ = [4muu¯dt+i(udu¯ u¯du)]=m(dt ωˆ ). (4.3)
J −1 4m2uu¯ − − P
−
Toreducethe numberofindependent Goldstonefields, similarlytothe flatspacecase[4],onemayimposethe
following conditions on the Cartan forms ω and ω¯ (inverse Higgs phenomenon [7]),
Z Z
λ λ u˙ +2imu
ω =0 ωˆ = i ωˆ =i ,
Z ⇒ Z − 1+λλ¯ P ⇒ 1+λλ¯ 1+4m2uu¯+2im(uu¯˙ u¯u˙)
−
λ¯ λ¯ u¯˙ 2imu¯
ω =0 ωˆ =i ωˆ = i − , (4.4)
Z ⇒ Z 1+λλ¯ P ⇒ 1+λλ¯ − 1+4m2uu¯+2im(uu¯˙ u¯u˙)
−
and, therefore,
(2mu+λ)(1+2muλ¯)
u˙ = i . (4.5)
− 1+λλ¯+2m(u¯λ+uλ¯)
Theseconstraintsarepurelykinematicones. Thus,torealizethisspontaneousbreakingofAdS U(1)symmetry
3
×
we need one complex scalar field, u(t) and u¯(t).
4
Using the constraints (4.4), one may further simplify the Cartan forms ω , ω (4.2) to be
P J
1 λλ¯ 1 λλ¯
ω = − ωˆ , ω =mdt m − ωˆ . (4.6)
P 1+λλ¯ P J − 1+λλ¯ P
Clearly, the simplest action, invariant under full AdS U(1) symmetry, is
3
×
2im(u˙u¯ uu¯˙) 2 u˙u¯˙
S = m ω = m dt 1+ − 4
0 − 0 P − 0 s 1 4m2uu¯ − (1 4m2uu¯)2
Z Z (cid:18) − (cid:19) −
1 λλ¯
= m dt − . (4.7)
− 0 1+λλ¯+2m(λu¯+λ¯u)
Z
One may check, that the curvature of the space with the metric
2im(duu¯ udu¯) 2 dudu¯
ds2 = dt+ − +4 (4.8)
− 1 4m2uu¯ (1 4m2uu¯)2
(cid:18) − (cid:19) −
is equal to = 6m2.
R −
KeepinginmindthattheCartanformω isshiftedbythefulltime derivativeunderalltransformationsofthe
3
AdS U(1) group, the invariant anyonic term, i.e. the action which results in the at most the third order time
3
×
derivativesequationsofmotionofthefieldsandwhichpossessesthe invarianceunderfullAdS U(1)symmetry,
3
×
acquires the form
λ˙λ¯ λ¯˙λ
S = ω =i dt − . (4.9)
anyon − 3 1 λλ¯
Z Z −
In terms of u, u¯ and their derivatives it reads
2i(u¨u¯˙ u¨¯u˙) 8mu˙u¯˙ +8im2(u˙u¯ u¯˙u)+4m(u¨u¯+u¨¯u)
S = dt − − −
anyon
Z  (1 4m2uu¯ 2im(u¯˙u u˙u¯))2 4u˙u¯˙
− − − −
q −1
1+4m2uu¯+2im(uu¯˙ u¯u˙)+ (1 4m2uu¯ 2im(u¯˙u u˙u¯))2 4u˙u¯˙ . (4.10)
×(cid:20) − q − − − − (cid:21) )
The actions S (4.7) and S (4.10) may be slightly simplified by passing to new variables q,q¯defined as
0 anyon
q =e2imtu, q¯=e−2imtu¯. (4.11)
In terms of these variables the action of the AdS particle reads
3
q˙q¯ q¯˙q 2 4q˙q¯˙
S = m dt 1 2im − , (4.12)
0 − 0 s − 1 4m2qq¯ − (1 4m2qq¯)2
Z (cid:18) − (cid:19) −
while the anyonic action (4.10) is simplified to be
2i(q¨q¯˙ q¨¯q˙)+8mq˙q¯˙
S = dt −
anyon
Z  (1 4m2qq¯+2im(q¯˙q q˙q¯))2 4q˙q¯˙
− − −
q −1
14m2qq¯+2im(q¯˙q q˙q¯)+ (1 4m2qq¯+2im(q¯˙q q˙q¯))2 4q˙q¯˙ . (4.13)
×(cid:20) − − q − − − (cid:21) )
This action can be rewritten through the Lagrangian of a particle on the pseudosphere and connection
PS2 µ
L A
as
q¨q¯˙ q¨¯q˙
2i − +8m
(1 4m2qq¯)2 LPS2
Sanyon = dt − . (4.14)
Z (1 q˙µ)2 4 1 q˙µ+ (1 q˙µ)2 4
µ PS2 µ µ PS2
−A − L −A −A − L
q (cid:18) q (cid:19)
5
Here
q˙q¯˙ q˙q¯ q¯˙q
= , q˙µ =2im − . (4.15)
LPS2 (1 4m2qq¯)2 Aµ 1 4m2qq¯
− −
The AdS anyon action, written in terms of the λ,λ¯ variables (4.9), has the same form as in the flat space-
3
time case [4]. This analogy is slightly broken for the AdS rigid particle action, which is invariant under the full
3
AdS U(1) symmetry and leads to equations of motion of at most fourth order in time derivatives,
3
×
S =β ωTω¯T = dt 1+2mλu¯+λ¯u 1+λλ¯ λ˙λ¯˙. (4.16)
rigid Z ωP Z (cid:18) 1+λλ¯ (cid:19) 1−λλ¯ 3!
The term proportional to m is needed to provide invariance with respect(cid:0)to the A(cid:1)dS symmetry, realized by left
3
multiplications of the coset element (4.1) as follows,
g =ei(aT+a¯T) δ t= iau¯ e2imt+1 +ia¯u e−2imt+1 , δ λ=a a¯λ2,
0 T T
⇒ − −
a 1
δ u= e(cid:0)−2imt 1 (cid:1)4m2uu¯(cid:0) a (cid:1)4m2a¯u2 , (4.17)
T
2m − − 2m −
g =ei(bZ+¯bZ) δ t= 2im be−2(cid:0)imtu¯ ¯be2im(cid:1)tu , (cid:0)δ u=b 1 (cid:1)4m2uu¯ e−2imt, δ λ=0.
0 Z Z Z
⇒ − − −
(cid:0) (cid:1) (cid:0) (cid:1)
Let us finally stress that the actions (4.7), (4.10) (4.16) we constructed are precisely the flat space-time
expressions when expressed in the terms of Cartan forms [4].
5 Fully supersymmetric case
Toconstructsupersymmetriccomponentactions,invariantunderbothunbrokenQandbrokenSsupersymmetries,
in full analogy with the flat case [4], one has to perform four steps:
Imposesomeadditionalconstraintstoreducethenumberofindependentsuperfieldsandimposeirreducibil-
•
ity constraints on the essential superfields;
Find the transformation properties of the physical components under both supersymmetries;
•
Write an ansatz for the component actions invariant under broken supersymmetries. The corresponding
• invariants are provided by the Cartan forms evaluated at θ =θ¯=0 condition;
Fix the arbitrary parameters in the ansatz by demanding invariance under the unbroken supersymmetry.
•
Let us go through these steps.
5.1 Irreducibility conditions
From the beginning, in our coset (3.1) there are three independent complex superfields u,ψ,λ (considering the
redefinitions (3.5)). To reduce the number of independent superfields we impose the same conditions (4.4) as in
the bosonic sector,
ΩZ =0 Ωˆ = i λ Ωˆ , Ωˆ =i λ¯ Ωˆ . (5.1)
ΩZ =0 ⇒ Z − 1+λλ¯ P Z 1+λλ¯ P
(cid:26)
Equating the coefficients of the differentials t,dθ and dθ¯we get
△
λ 1 4m2uu¯
1 4mψψ¯ u= 2imu i − ,
− ∇t − − 1+λλ¯+2m λu¯+λ¯u
(cid:0) (cid:1)
(cid:0)1 4mψψ¯(cid:1) u¯ =2imu¯+i λ¯ 1−4m2(cid:0)uu¯ (cid:1) (5.2)
− ∇t 1+λλ¯+2m λu¯+λ¯u
(cid:0) (cid:1)
(cid:0) (cid:1)
and (cid:0) (cid:1)
u+4muψ u=0, u¯ 4mu¯ψ¯ u¯ =0, (5.3)
t t
∇ ∇ ∇ − ∇
u= 2iψ¯ 1 4m2uu¯+2imu¯ u , u¯ = 2iψ 1 4m2uu¯+2imu u¯ . (5.4)
t t
∇ − − ∇ ∇ − − ∇
(cid:0) (cid:1) (cid:0) (cid:1)
6
These relations simplify the form Ω to
P
1 λλ¯ t+4m uψdθ u¯ψ¯dθ¯
Ω = − Ωˆ , Ωˆ = 1+4mψψ¯ △ − . (5.5)
P 1+λλ¯ P P " 1+(cid:0)2mu¯λ+u¯λ¯ (cid:1)#
1+λλ
(cid:0) (cid:1)
Inprinciple,(5.2),(5.3),(5.4)solvealltasks. Indeed,using(5.2)onemayexpressthesuperfieldsλ,λ¯ interms
of time derivatives of u,u¯, while (5.3) can be solved to express the fermionic superfields ψ,ψ¯ in terms of spinor
covariant derivatives of the same u,u¯. Thus, like in the flat case [4], we remain with only one N = 2 complex
bosonicsuperfieldu(t,θ,θ¯),restrictedby(5.3)tobecovariantlychiral,withslightlymodifiedchiralityconditions.
However, in what follows we are going to use as independent components the θ = θ¯ = 0 projections of the
superfields ψ,ψ¯ instead of the projections of u and u¯. Therefore, it would be useful to find the consequences
∇ ∇
of the constraints (5.2), (5.3), (5.4).
Firstofall,actingby onthe firstequationin(5.3)andby onthesecondoneandusingthe algebra(3.13)
∇ ∇
of the covariant derivatives, we get the conditions
ψ+4muψ ψ =0, ψ¯ 4mu¯ψ¯ ψ¯ =0. (5.6)
t t
∇ ∇ ∇ − ∇
Note, that this asserts the self-consistency of the modified chirality constraints (5.3) because
+4muψ , +4muψ =0, 4mu¯ψ¯ , 4mu¯ψ¯ =0. (5.7)
t t t t
{∇ ∇ ∇ ∇ } ∇− ∇ ∇− ∇
(cid:8) (cid:9)
Secondly,actingby onthefirstequationin(5.3)andby onthefirstequationin(5.4)andaddingthe results,
∇ ∇
after quite lengthly calculations with heavy use of (3.13), we obtain
λ+2mu
ψ¯ = i 1 4mψψ¯ 8im2uψψ¯ 4muψ ψ¯. (5.8)
∇ − 1+2mu¯λ − − − ∇t
(cid:0) (cid:1)
Repeating similar calculations with the second equations in (5.3), (5.4) yields the conjugated expression
λ¯+2mu¯
ψ =i 1 4mψψ¯ +8im2u¯ψψ¯ +4mu¯ψ¯ ψ. (5.9)
∇ 1+2muλ¯ − ∇t
(cid:0) (cid:1)
Now we have all ingredients needed for constructing the component action.
Before closing this subsection let us visualize a more simple way to obtain (5.8) and (5.9). The idea consists
in the using the constraints2
Ω =0, Ω =0. (5.10)
S|ΩQ,ΩQ S|ΩQ,ΩQ
Here, the notation means that the Cartan forms Ω and Ω must be expanded in the forms Ω ,Ω ,Ω
|ΩQ,ΩQ S S P Q Q
before nullifying their -projections. At first sight these conditions seem to be more complicated due to the
|ΩQ,ΩQ
highly nontrivial structure of the Cartan forms involved. However, this is not the case and the calculations can
be simplified using the following procedure.
With the help of our definitions (3.7), the first constraint in (5.10) can be formally represented as
Ωˆ iλ¯Ωˆ
Ω = S − Q =Ω X (5.11)
S P
1 λλ¯
−
where X is some expression which is defined by tphis equation. The main difference between (5.10) and (5.11)
is that the latter one is written as the equation on forms. Therefore, one may just substitute in (5.11) the
exact expressions from (3.8) and equate on both sides the coefficients of the differentials t,dθ and dθ¯. The t
coefficient relation yields X in terms of ψ. Substituting this expression for X in the△dθ,dθ¯ -projections△of
t
∇
(5.11)weimmediatelyobtainthe firstequationin(5.6)andalso(5.9). The sameprocedureappliedtothe second
constraint in (5.10) produces the second equation in (5.6) as well as (5.8).
These considerationsdemonstrate that the constraints (5.10) are a consequence of our basic constraints (5.1).
2Theseconditions,beingsomevariantofthesuperembeddingconditions [18],weretrivialintheflatcase[4].
7
5.2 Transformation properties of the components
Aswearegoingtoconstructthecomponentactions,weneedtoknowthetransformationlawsforthecomponents.
We denote the components of the superfields in the following way,
u =u, u¯ =u¯, ψ =ψ, ψ¯ =ψ¯, λ =λ, λ¯ =λ¯. (5.12)
θ=0 θ=0 θ=0 θ=0 θ=0 θ=0
| | | | | |
The equations (5.2) evaluated at θ =θ¯=0 provide relations between λ,λ¯ and the time derivatives of u,u¯. Thus,
λ,λ¯ are not independent components. We introduce these variables just to simplify many expressions in what
follows.
Broken S supersymmetry
The transformationpropertiesof our components (5.12) under brokensupersymmetry can be easily learnedfrom
(3.3). Before listing these transformations, we point out that, in contrast to the flat case [4], the superspace
coordinates θ and θ¯are not invariant under the broken supersymmetry (3.3),
δ θ =4mεe2imtψ¯θ, δ θ¯= 4mε¯e−2imtψθ¯.
S S
−
However, the right hand sides of these variations disappear in the limit θ = θ¯ = 0 and, thus, the set of the
components u,u¯,ψ,ψ¯ is closed under the broken supersymmetry. The corresponding transformations read
(cid:8) δ t(cid:9)=i εe2imtψ¯+ε¯e−2imtψ ,
S
δSψ =ε(cid:0)e2imt 1+2mψψ¯ , δ(cid:1)Sψ¯=ε¯e−2imt 1+2mψψ¯ , δSu=0, δSu¯=0. (5.13)
It is rather easy to check that the(cid:0)expression(cid:1) (cid:0) (cid:1)
1+4mψψ¯ t = 1+4mψψ¯ dt i ψdψ¯+ψ¯dψ dt (5.14)
θ=0
△ | − ≡E
is invariant with respect t(cid:0)o (5.13). T(cid:1)herefore, it(cid:0)is natural (cid:1)to(cid:2)define(cid:0)a new covar(cid:1)i(cid:3)antderivative as
= −1∂ , −1 = 1 4mψψ¯ 1 i ψψ¯+ ψ¯ψ . (5.15)
t t t t
D E E − − D D
It then immediately follows from (5.14) and (5.15(cid:0)) that (cid:1)(cid:2) (cid:0) (cid:1)(cid:3)
δ u=0, δ u¯=0 δ λ=δ λ¯ =0. (5.16)
S t S t S S
D D ⇒
Unbroken Q supersymmetry
The transformations under unbroken Q-supersymmetry can be defined in a usual way as
δ f = ǫD+¯ǫD f = ǫ +ǫ¯ f H∂ ,
Q θ→0 θ→0 t
− | − ∇ ∇ | −
H = iǫ(cid:0)ψ ψ¯ +ψ¯(cid:1) ψ θ→0+(cid:0) iǫ¯ ψ ψ¯(cid:1)+ψ¯ ψ θ→0. (5.17)
∇ ∇ | ∇ ∇ |
For example, (cid:0) (cid:1) (cid:0) (cid:1)
δ u=2iǫ¯ψ¯ 1 4m2uu¯ +4m ǫuψ ǫ¯u¯ψ¯ u Hu˙, δ ψ = ǫ ψ+ǫ¯ ψ Hψ˙. (5.18)
Q t Q θ→0
− − D − − ∇ ∇ | −
Another important(cid:0)object is th(cid:1)e vielbe(cid:0)in (5.15)(cid:1)which transforms as fol(cid:0)lows (cid:1)
E
δ = 2i 1+4mψψ¯ ǫ ψ+ǫ¯ ψ ψ¯˙ + ǫ ψ¯ +ǫ¯ ψ¯ ψ˙
Q θ→0 θ→0
E E ∇ ∇ | ∇ ∇ |
4(cid:0)m ǫ ψ+(cid:1) ǫ¯h(cid:0)ψ ψ¯+4(cid:1)m ǫ ψ¯ (cid:0)+ǫ¯ ψ¯ ψ(cid:1) ∂ Hi . (5.19)
θ→0 θ→0 t
− E ∇ ∇ | E ∇ ∇ | − E
Of course, to find the explicit form(cid:0)of the transf(cid:1)ormations (5.1(cid:0)9) one has to(cid:1) use the re(cid:0)latio(cid:1)ns (5.6), (5.8), (5.9)
evaluated at θ =θ¯=0:
( ψ) +4muψ ψ =0, ψ¯ 4mu¯ψ¯ ψ¯=0,
θ=0 t θ=0 t
∇ | D ∇ | − D
λ+2mu
ψ¯ = i 1 (cid:0)4mψ(cid:1)ψ¯ 8im2uψψ¯ 4muψ ψ¯,
∇ |θ=0 − 1+2mu¯λ − − − Dt
(cid:0) (cid:1) λ¯+2mu¯ (cid:0) (cid:1)
ψ =i 1 4mψψ¯ +8im2u¯ψψ¯+4mu¯ψ¯ ψ. (5.20)
∇ |θ=0 1+2muλ¯ − Dt
In particular, the transform(cid:0)atio(cid:1)n (5.19) acquires th(cid:0)e form (cid:1)
λ+2mu λ¯+2mu¯
δ = ∂ H +2 ǫ ψ 2imψ 2 ǫ¯ ψ¯+2imψ¯ . (5.21)
QE − t E E1+2mu¯λ Dt − − E1+2muλ¯ Dt
Finally, we stress that the re(cid:0)latio(cid:1)ns between the co(cid:0)mponents u,(cid:1)u¯ and λ,λ¯ are giv(cid:0)enby the foll(cid:1)owing expressions,
λ 1 4m2uu¯ λ¯ 1 4m2uu¯
u= 2imu i − , u¯=2imu¯+i − (5.22)
Dt − − 1+λλ¯+2m λu¯+λ¯u Dt 1+λλ¯+2m λu¯+λ¯u
(cid:0) (cid:1) (cid:0) (cid:1)
(cid:0) (cid:1) (cid:0) (cid:1)
8
5.3 Actions
Wearereadytoconstructthesupersymmetricgeneralizationoftheactions(4.7)and(4.9). Astheyhavedifferent
dimension, these actions must be invariant individually.
Superparticle
It is easy to check that the evident ansatz
dt F(u,u¯,λ,λ¯)
E
Z
is invariant under the broken supersymmetry for any function F because, in virtue of (5.13), (5.14), (5.16),
δ F =0, δ (dt )=0. (5.23)
S S
E
The desired bosonic limit (4.7) immediately fixes the function F up to constant α
1 λλ¯
S = m dt α+ − . (5.24)
0 − 0 E" 1+λλ¯+2m uλ¯+u¯λ #
Z
(cid:0) (cid:1)
This constant α can be determined as unity either from linearized Q supersymmetry invariance or from the flat
space-time action of [4]. Let us explicitly demonstrate that the action
1 λλ¯
S = m dt 1+ − m dt (5.25)
0 − 0 E" 1+λλ¯+2m uλ¯+u¯λ #≡− 0 L
Z Z
(cid:0) (cid:1)
is invariant under the unbroken Q supersymmetry.
Using (5.18) and (5.22), one finds that
1+2mu¯λ
δ λ= 2ǫ¯ ψ¯+2imψ¯ 1+λλ¯+2m(uλ¯ u¯λ) +4m ǫuψ ǫ¯u¯ψ¯ λ H∂ λ, δ λ¯ =(δ λ)†. (5.26)
Q − Dt 1+2muλ¯ − − Dt − t Q Q
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Now, the variation of integrand in (5.25) reads
1 λλ¯
δ = ∂ H +4m ǫuψ ǫ¯u¯ψ¯ −
QL − t L E − Dt"1+λλ¯+2m uλ¯+u¯λ #
(cid:0) (cid:1) (cid:0) (cid:1)
+4m ǫ ψu ǫ¯ ψ¯u¯ 1−(cid:0)λλ¯ (cid:1)
E Dt − Dt 1+λλ¯+2m uλ¯+u¯λ
4im ǫψ 1−λλ¯ 1+2muλ¯ λ+2mu) (cid:0)4im ǫ¯ψ¯ 1−λλ¯(cid:1) 1+2mu¯λ (cid:0)λ¯+2mu¯)(cid:1). (5.27)
− E (cid:0) 1+λ(cid:1)(cid:0)λ¯+2m uλ¯(cid:1)+(cid:0) u¯λ 2 − E (cid:0) 1+λ(cid:1)(cid:0)λ¯+2m uλ¯(cid:1)+(cid:0) u¯λ 2
Using the relations (5.22(cid:2)), the last line(cid:0)in (5.27)(cid:1)(cid:3)may be represente(cid:2)d as (cid:0) (cid:1)(cid:3)
1 λλ¯
4m ǫψ u ǫ¯ ψ¯ u¯ − ,
E Dt − Dt Dt 1+λλ¯+2m uλ¯+u¯λ
(cid:0) (cid:1)
and, therefore, (cid:0) (cid:1)
ǫψu ǫ¯ψ¯u¯ 1 λλ¯ ǫψu ǫ¯ψ¯u¯ 1 λλ¯
δ = ∂ H +4m − − =∂ H +4m − − . (5.28)
QL − t L EDt"1(cid:0)+λλ¯+2m(cid:1)(cid:0)uλ¯+u¯λ(cid:1) # t"− L 1(cid:0)+λλ¯+2m(cid:1)(cid:0)uλ¯+u¯λ(cid:1) #
(cid:0) (cid:1)
(cid:0) (cid:1) (cid:0) (cid:1)
Thus, the action (5.25) is invariant under both the broken S and unbroken Q supersymmetries, and it is the
action of the N =(2,0) AdS superparticle.
3
The AdS superparticleaction (5.25) may be written in terms of the Cartanforms evaluatedat θ =dθ =0 in
3
a rather simple way as
m
S = 0 Ω . (5.29)
0 m J|θ=0
Z
9